Equation of a Barotropic Fluid on a Rotating Spherical Surface and its Inertial Manifold

Abstract

The inertial manifold is a positive invariant set which exponentially attracts all the trajectories of a dissipative dynamic system. It was introduced for the purpose of studying the asymptotic behaviour of such systems. The initial infinitely dimensional dynamic system, generated by a partial differential evolution equation, can be projected on to it, in order to obtain the final system of ordinary differential equations (inertial equations). These equations then simulate the initial object. Although the inertial manifold is an object relatively simpler than the attractor (a very complicated set of non-integer dimension may be an attractor) it is more difficult to prove its existence than that of the attractor. The equation of a barotropic fluid on a rotating spherical surface is one of the examples of dissipative dynamic systems with an inertial manifold. This kindles the hope that also the equations of the dynamics of the real atmosphere will have an inertial manifold. The reduction of the sample system to this Lipschitz manifold of finite dimension thus justifies us in analysing the behaviour of the atmosphere on non-linear models of finite dimensions and few parameters, in a finite system of ordinary differential equations.